Question

Assertion: If ABCD is a rhombus whose one angle is 60°, then the ratio of the lengths of its diagonals is $\sqrt{3}:1$.
Reason: Median of triangle divides it into two triangles of equal area.
(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.

Answer 1

(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.

Reason (R) is clearly true.
The explanation of assertion (A)​ is as follows:
ABCD is a rhombus. So, all of its sides are equal.
Now, BC = DC
⇒∠BDC = ∠DBC = x$°$
Also, ∠BCD = 60$°$
∴ x$°$ + x$°$ + 60$°$ = 180$°$
⇒​2x$°$ = 120$°$
⇒​ x$°$ = 60$°$
∴ ∠BCD = ∠BDC = ∠DBC = 60$°$
So, ​∆BCD is an equilateral triangle.
i.e., BD = BC = a
∴ OB = $\frac{a}{2}$
Now, in ∆ OAB, we have:
$$\begin{gathered} O{A}^2=A{B}^2-O{B}^2=a^2-{\left(\frac{a}{2}\right)}^2=\frac{3a^2}{4} \\ \Rightarrow OA=\frac{\sqrt{3}a}{2} \\ \Rightarrow AC=2\times \frac{\sqrt{3}a}{2}=\sqrt{3}a \end{gathered}$$
$\therefore AC:BD=\sqrt{3}a:a=\sqrt{3}:1$
Thus, assertion (A)​ is also true, but reason (R) does not give (A).
​Hence, the correct answer is (b).